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Properties

Label 1-4176-4176.3091-r0-0-0
Degree $1$
Conductor $4176$
Sign $0.368 - 0.929i$
Analytic cond. $19.3932$
Root an. cond. $19.3932$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)5-s + (−0.5 + 0.866i)7-s + (−0.5 + 0.866i)11-s + (−0.866 + 0.5i)13-s − i·17-s − 19-s + (−0.5 − 0.866i)23-s + (0.5 − 0.866i)25-s + (0.866 − 0.5i)31-s − i·35-s + 37-s + (−0.866 + 0.5i)41-s + (−0.5 + 0.866i)43-s + (0.866 + 0.5i)47-s + (−0.5 − 0.866i)49-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)5-s + (−0.5 + 0.866i)7-s + (−0.5 + 0.866i)11-s + (−0.866 + 0.5i)13-s − i·17-s − 19-s + (−0.5 − 0.866i)23-s + (0.5 − 0.866i)25-s + (0.866 − 0.5i)31-s − i·35-s + 37-s + (−0.866 + 0.5i)41-s + (−0.5 + 0.866i)43-s + (0.866 + 0.5i)47-s + (−0.5 − 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4176 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.368 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4176 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.368 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4176\)    =    \(2^{4} \cdot 3^{2} \cdot 29\)
Sign: $0.368 - 0.929i$
Analytic conductor: \(19.3932\)
Root analytic conductor: \(19.3932\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4176} (3091, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4176,\ (0:\ ),\ 0.368 - 0.929i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9733622594 - 0.6615050594i\)
\(L(\frac12)\) \(\approx\) \(0.9733622594 - 0.6615050594i\)
\(L(1)\) \(\approx\) \(0.9842312799 - 0.04317046567i\)
\(L(1)\) \(\approx\) \(0.9842312799 - 0.04317046567i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
29 \( 1 \)
good5 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.866 + 0.5i)T \)
17 \( 1 - iT \)
19 \( 1 - T \)
23 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (0.866 - 0.5i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.866 + 0.5i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 - iT \)
59 \( 1 + (0.866 - 0.5i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 - T \)
73 \( 1 - iT \)
79 \( 1 + (-0.866 - 0.5i)T \)
83 \( 1 + (0.866 + 0.5i)T \)
89 \( 1 - iT \)
97 \( 1 + (-0.866 - 0.5i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.66534141991527286413188382432, −17.63801443961390118768059616912, −17.17672811187652836158328627152, −16.76994872706074756480138462039, −15.76236728198016144667082358643, −15.130879443950154413470817424047, −14.35975182700162053993655877761, −13.71110858895802468185763629441, −13.197789188521509656105141463047, −12.603088646935612420893744135802, −11.62300544214051577884039780863, −10.63257883666427805597709108060, −10.376755665426006724676245984873, −9.81545914364994730530953708210, −8.847985518958847335842744999896, −8.08638180366689725333123754464, −7.301816656159597054294212267077, −6.584944282269727372724300432394, −5.92114415029593958493830480202, −5.30507453276863836838681690660, −4.23768771293718632964262619263, −3.46420285102455715106406245609, −2.72698749137170020692393930762, −1.94450649865599886231141200027, −0.880407849735071642641018979529, 0.35877588561300903288134204334, 1.76191870401533177254888168817, 2.40369625990651591970680195012, 2.84928800083186397542959886317, 4.38625046572707252489828343265, 4.76660702132037889061467914174, 5.57986168357639878019822515728, 6.36781046425554793540998415547, 6.89458085082252672147293041521, 7.96471175128072418330204152139, 8.638600251595509742267144483831, 9.55178958808854818668812193897, 9.7299778834434619302933121038, 10.507139996642966265155731833980, 11.66087264481244413134073720588, 12.19562870157528813856894617547, 12.85213581367257132111446924858, 13.333497485085877467462321733889, 14.26390744325639760342278043225, 14.88204210248158016821703223665, 15.52658528846858535894414828097, 16.46356758735477129604138460623, 16.73695324540030630344041455465, 17.77675084170171595550357614846, 18.10194748629529811412823941046

Graph of the $Z$-function along the critical line